If certain moves are compulsory, will there still be a need for a quiescence search?
I understand your question to be:


  If some moves are compulsory, and my agent has no choice about which move to make next, do I need to perform a search, or can I just return the compulsory move?


The answer depends on what your goal is.

If your goal is to make an interactive agent that will play the game against you, then you are correct: there's no need to perform a search. Just return the compulsory move, and then run the search next time your agent has a choice about what to do.

If your goal is to determine the optimal way to play a game, or the expected payoff from a certain game position (another common use of search techniques), then you should run the search as normal, since the forced move won't necessarily lead to a particular end state.

Tangentially, if you're interested in ways to speed up search for checkers, check out the Chinook papers. There's a popularized account here, and more technical ones here and here by Schaeffer et al.